YUMMyYonsei Unstructured Moving Mesh

Hydrodynamic Simulation code

Kiyun Yun1, Suk‐Jin Yoon1,2, Juhan Kim3, and Sungsoo Kim41 Center for Galaxy Evolution Research, Yonsei University

2Department of Astronomy, Yonsei University3Korea Institute for Advanced Study

4Department of Astronomy & Space Science, Kyung Hee University

YUMMy | KNAG | 07/03/2015

Outline Introduction to the Unstructured Moving Mesh

UnstructuredMovingMesh

Test1D Shock tube test Interacting blast wave testGresho VertexKelvin-Helmholtz InstabilityRayleigh-Taylor InstabilityNoh Shock TestMoving Boundary TestWind Tunnel

YUMMy | KNAG | 07/03/2015

2 Hydrodynamic descriptionsLagrangian

A way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. (Wikipedia)

Smoothed-particle hydrodynamics(SPH); GASOLINE, GADGET-2,…pro. : easy to use, analyze, Galilean invariance / con. : sticky, instability,

dissipation

EulerianA way of looking at fluid motion that focused on specific locations in

the space through which the fluid flows as time passes. (Wikipedia)Cartesian Grid, Adaptive Mesh Refinement(AMR) ; ENZO, FLASH,

ATHENA, RAMSES, PLUTO,…pro. : accurate, instability / con. : Galilean invariance, coordinate

Hybird : Lagrangian + Eulerian AREPO, GIZMO, YUMMy, …pro. : ?? / con. : ??

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Unstructured Moving MeshBased on both of Lagrangian and Eulerian

Lagrangian Eulerian

YUMMy is

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Unstructured Moving Mesh Irregular shape; Voronoi Cell

∈ | , ,

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Unstructured Moving Mesh Irregular shape; Voronoi Cell

Each mesh has more surfaces(>5 in 2D, >10 in 3D) which face arbitrary direction than the cube used in conventional code →more accurate

hydro-quantities evolution.

Voronoi Cell Voronoi Cells

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Unstructured Moving Mesh Voronoi tessellation

Incremental Expanding Method(IEM)• Intuitive, fast, extendable to the high dimension(>3)

Hybrid-Neighbor Searching Method• Tree + ‘TreeMap’ : reduce required total number of searching cubes• ~2 times faster than just used in tree algorithm

Speed ~

We propose the robust solution to deal with the degeneracy problem• min ⋅

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Unstructured Moving Mesh Meshes move according to the speed → Galilean-invariance

of flow mostly derived by Euler equations for regularizing the shape mesh

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Unstructured Moving MeshEulerian Description

Godunov’s 2nd order methodMUSCL-Hancock SchemeFor the 2nd order accuracy

• Barth and Jesperson TVD for unstructured mesh• Q-R Factorization : the slope in the mesh

HLLC Approximate Riemann solver• Adaptive Noniterative Riemann Solver

• Two Rarefaction Riemann Solver(TRRS)• Two Shock Riemann Solver(TSRS)• Primitive Variable Riemann Solver(PVRS)

⋅ 0

, 

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TEST 1D

1. 1D Riemann Problem : Rarefication‐Shock Test2. Interacting Blast Waves

2D3. Gresh Vertex4. Kelvin‐Helmholtz Instability5. Ray‐Taylor Instability6. Noh Shock Test7. Moving Boundary Test8. Wind Tunnel Test

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AREPO

1D Shock‐Tube Test

Springel, 2009

YUMMy5.0

1 2 3 4 5 6 7 8

• 0 : 1,  1, 0

• 0 :  0.1795,  0.25,0

• 1.4

Initial Conditions

YUMMy | KNAG | 07/03/2015

Springel, 2009

AREPO

• 1,  1.4,  ∈ 0, 1

• 1000 for  0.1,  100 for  0.9,  0.01 for elsewhereInitial Conditions

0.038

Interacting blast waves(Woodward & Collella, 1984)   Moving Mesh 1 2 3 4 5 6 7 8

YUMMy

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AREPO

0.038

YUMMy

Interacting blast waves(Woodward & Collella, 1984)   Static Mesh 1 2 3 4 5 6 7 8

Springel, 2009

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YUMMy

Interacting blast waves(Woodward & Collella, 1984) 1 2 3 4 5 6 7 8

0.038

Comparison between moving and static mesh with same resolution

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• Azimuthal velocity         5 0 0.2

2 5 0.2 0.40 0.4

•5 25 2⁄ 0 0.2

9 25 2⁄ 20 4 ln 0.2⁄ 0.2 0.43 4 ln 2 0.4

,    1

0

Initial Conditions

Gresho Vortex 1 2 3 4 5 6 7 8

Frame velocity,   1

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Springel, 2009

AREPO

Gresho Vortex,  with Moving Mesh 1 2 3 4 5 6 7 8

YUMMy

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Springel, 2009

AREPO

Gresho Vortex,  , with Moving Mesh 1 2 3 4 5 6 7 8

YUMMy

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Springel, 2009

Gresho Vortex,  , with Static Mesh 1 2 3 4 5 6 7 8

AREPO(static mesh)

ATHENA

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1024

Periodic boundary

∈ 0,1 , ∈ 0,1

2, 0.5 for 0.5 0.25

1, 0.5 for Elsewhere

2.5, 5/3

,sin 4 exp .

exp .

Initial Conditions

Kelvin‐Helmholtz Instability 1 2 3 4 5 6 7 8

YUMMy

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Side-by-Side Comparison

y

Kelvin‐Helmholtz Instability  Moving Mesh 1 2 3 4 5 6 7 8

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Springel, 2009AREPO

Kelvin‐Helmholtz Instability  Moving Mesh 1 2 3 4 5 6 7 8

YUMMy

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Springel, 2009

AREPO

Kelvin‐Helmholtz Instability  Static Mesh 1 2 3 4 5 6 7 8

YUMMy

ATHENA

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?

?

Kelvin‐Helmholtz Instability  Galilean Invariance  , 1 2 3 4 5 6 7 8

, 0,0, 1,1

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∈ 0,0.5 , ∈ 0,1.5 Periodic boundary at x-boundaries, reflecting wall at top and bottom

TOP : 2 BOTTOM : 1 2.5, 1.4 0.75 , 0.1 0, everywhere , 1 cos 4 1 cos 4 3⁄ , 0.0025

Initial Conditions

Springel, 2009

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

YUMMy | KNAG | 07/03/2015

0 1 10

48 144

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

0 1 10

Various Frame Velocities, Moving Mesh

15.0

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0 1 10

48 144

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

0 1 10

Various Frame Velocities, Static Mesh

15.0

• Results are influenced by the frame velocity• No Galilean invariance

YUMMy | KNAG | 07/03/2015

0 1 10

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

0 1 10

YUMMyAREPO

AREPO vs YUMMy for various frame velocities and static mesh

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Galilean invariance

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

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Galilean invariance

Springel, 2009

10

1

AREPO  0, 1, 10

ATHENA

ATHENA

ATHENA

0

Ray‐Taylor Instability 1 2 3 4 5 6 7 8

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Galilean invariance

The laws of motion are the same in all inertial frames.

Any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary.(wiki)

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Galilean invariance

Hydro‐quantities( , , ) projected by the direction of surfaces are averaged in the adjacent cells

Cartesian Grids

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Noh Shock Test 1 2 3 4 5 6 7 8

2.0

YUMMy

AREPO

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Moving Boundary Test 1 2 3 4 5 6 7 8

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2(multiple) objects

Ma = 2.7

Wind Tunnel 1 2 3 4 5 6 7 8

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Wing

Ma = 1.5

• Easy/Possible to set arbitrary shape of boundary condition• Cartesian coordinate, without needing different coordinates (polar, cylindrical)

50 , 200 for medium, 2000 for boundary condition

50 200

Wind Tunnel 1 2 3 4 5 6 7 8

YUMMy | KNAG | 07/03/2015

Characters(very complex boundary)

Ma = 1.5600 for medium, 2000 for boundary condition

Wind Tunnel 1 2 3 4 5 6 7 8

• Karman vertox street• Turbulences• Bow shock

YUMMy | KNAG | 07/03/2015

Summary YUMMy; Yonsei Unstructured Moving Mesh Hydrodynamic Simulation code

Both of Lagrangian and Eulerian description are implemented.

The Voronoi Tessellation is adapted for unstructured mesh. The IEM, Hybrid-Neighbor Searching and schemes are implemented → high performance

Figuring out the robust solution to generate voronoi tessellation → in process of publishing paper

YUMMy is the Galilean-invariance code. Suitable to implement for cosmological simulation and interaction between multiple

objects

The results of both static and moving mesh tests are consistent with that of another moving mesh code, AREPO.

It’s possible to use arbitrary shape of objects without needing different coordinates (polar, cylindrical)